Showing papers on "Fuzzy number published in 1986"
TL;DR: Various properties are proved, which are connected to the operations and relations over sets, and with modal and topological operators, defined over the set of IFS's.
13,376 citations
TL;DR: The new definition of expectation generalizes the integral of a set-valued function and derives the Lebesgue-dominated convergence type theorem by considering a suitable generalization of the Hausdorff metric.
1,814 citations
TL;DR: This paper shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in [4] to define differentiation and integration of fuzzy-valued functions in ways that parallel closely the corresponding definitions for real differentiation and Integration.
1,273 citations
TL;DR: It is shown that the proposed representation exists for certain families of the conjugate pairs of t-norms and t-conorms and resolves some of the difficulties associated with particular interpretations of conjunction, disjuntion, and implication in fuzzy set theories.
1,041 citations
Book•
01 Mar 1986679 citations
TL;DR: A new nonprobabilistic entropy measure is introduced in the context of fuzzy sets or messages and the theory of subsethood is shown to solve one of the major problems with Bayes-theorem learning and its variants—the problem of requiring that the space of alternatives be partitioned into disjoint exhaustive hypotheses.
605 citations
01 Mar 1986
TL;DR: Fuzzy set theory has a number of properties that make it suitable for formalizing the uncertain information upon which medical diagnosis and treatment is usually based, and trials performed with the medical expert system CADIAG-2 suggest that it might be a suitable basis for the development of a computerized diagnosis system.
Abstract: Fuzzy set theory has a number of properties that make it suitable for formalizing the uncertain information upon which medical diagnosis and treatment is usually based. Firstly, it defines inexact medical entities as fuzzy sets. Secondly, it provides a linguistic approach with an excellent approximation to texts. Finally, fuzzy logic offers reasoning methods capable of drawing approximate inferences. These facts suggest that fuzzy set theory might be a suitable basis for the development of a computerized diagnosis system. This is verified by trials performed with the medical expert system CADIAG-2, which uses fuzzy set theory to formalize medical relationships and fuzzy logic to model the diagnostic process.
392 citations
TL;DR: It is shown that the method is capable of generating membership functions in accordance with the possibility-probability consistency principle for fuzzy sets whose elements have a defining feature with a known probability density function in the universe of discourse.
333 citations
TL;DR: A strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables, whose values are fuzzy sets with compact levels.
Abstract: A strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables, whose values are fuzzy sets with compact levels. The proofs are based on embedding theorems as well as on probability techniques in Banach space.
315 citations
TL;DR: Fuzzy set theory is established as a theoretical basis for ordination, and is employed in a sequence of examples in an analysis of forest vegetation of western Montana, U.S.A.
Abstract: Fuzzy set theory is an extension of classical set theory where elements of a set have grades of membership ranging from zero for non-membership to one for full membership. Exactly as for classical sets, there exist operators, relations, and mappings appropriate for these fuzzy sets. This paper presents the concepts of fuzzy sets, operations, relations, and mappings in an ecological context. Fuzzy set theory is then established as a theoretical basis for ordination, and is employed in a sequence of examples in an analysis of forest vegetation of western Montana, U.S.A. The example ordinations show how site characteristics can be analyzed for their effect on vegetation composition, and how different site factors can be synthesized into complex environmental factors using the calculus of fuzzy set theory. In contrast to current ordination methods, ordinations based on fuzzy set theory require the investigator to hypothesize an ecological relationship between vegetation and environment, or between different vegatation compositions, before constructing the ordination. The plotted ordination is then viewed as evidence to corroborate or discredit the hypothesis. I am grateful to Dr R. D. Pfister (formerly USDA Forest Service) for kind permission to publish data from a Forest Service study. I would like to gratefully acknowledge the helpful comments and criticisms of Drs. G. Cottam, J. D. Aber, T. F. H. Allen, E. W. Beals, I. C. Prentice, C. G. Lorimer, and two anonymous reviewers.
258 citations
TL;DR: This method for solving a multicriteria linear program where the coefficients of the objective functions and the constraints are fuzzy numbers of the L-R type is elaborated with a view to the application to the development of a water supply system.
TL;DR: Some fundamental properties of fuzzy binary relations and certain conditions of reasonable orderings of fuzzy Utilities are introduced and a method for constructing a fuzzy preference relation on a given set of fuzzy utilities is proposed for the sake of rational decision making.
01 Jan 1986
TL;DR: This paper presents an outline of a theory of usuality based on fuzzy logic and applications of fuzzy subsets theory and mathematical programming, and some particular applications.
Abstract: 1: Some theoretical Aspects.- 1.1 Mathematics and fuzziness.- 1.2 Radon-Nikodym Theorem for fuzzy set-valued measures.- 1.3 Construction of a probability distribution from a fuzzy information.- 1.4 Convolution of fuzzyness and probability.- 1.5 Fuzzy sets and subobjects.- 2: From theory to applications.- 2.1 Outline of a theory of usuality based on fuzzy logic.- 2.2 Fuzzy sets theory and mathematical programming.- 2.3 Decisions with usual values.- 2.4 Support logic programming.- 2.5 Hybrid data - various associations between fuzzy subsets and random variables.- 2.6 Fuzzy relation equations : methodology and applications.- 3: Various particular applications.- 3.1 Multi criteria decision making in crisp and fuzzy environments.- 3.2 Fuzzy subsets applications in O.R. and management.- 3.3 Character recognition by means of fuzzy set reasoning.- 3.4 Computerized electrocardiography and fuzzy sets.- 3.5 Medical applications with fuzzy sets.- 3.6 Fuzzy subsets in didactic processes.
TL;DR: It is shown that Sugeno's fuzzy integral is a particular case of them and they are studied to study their principal properties.
TL;DR: All four designs yield comparable (usually within 4%) error rates; the Fuzzy c-Means (FCM) based k-NNR is usually the best design; the FCM/1-NPR is the most efficient and perhaps most useful of the four designs; and finally, that generalized NNR's are an important and useful extension of the conventional ones.
TL;DR: The concept of risk evaluation, using linguistic representation of the likelihood of the occurrence of a hazardous event, exposure, and possible consequences of that event, and the approximate reasoning technique based on fuzzy logic is used to derive fuzzy values of risk.
Book•
22 Dec 1986
TL;DR: Fuzzy Set Theory and Nonlinear Models: From Words to Numbers to Numbers and Back Again.
Abstract: Introduction: Why Fuzzy Sets?.- 1. Fuzzy Set Theory: The Basics.- 2. Is Fuzzy Set Theory Realistic?.- 3. Fuzzy Scales and Measurement.- 4. Fuzziness and Internal Category Structure.- 5. Intercategory Relations and Taxonomic Analysis.- 6. Fuzzy Set Theory and Nonlinear Models.- 7. Prediction and Fuzzy Logic.- Epilogue: From Words to Numbers and Back Again.- Technical Glossary.- References.- Author Index.
TL;DR: This paper shows how to solve problems using probability theory that the fuzzy approaches claim probability cannot solve by using the view that probabilities are a measure of belief in a proposition in a particular context, limitations imposed by the frequency interpretations of probability are avoided.
Abstract: This paper shows how to solve problems using probability theory that the fuzzy approaches claim probability cannot solve. By using the view that probabilities are a measure of belief in a proposition in a particular context, limitations imposed by the frequncy interpretations of probability are avoided. The various fuzzy approaches (fuzzy sets, fuzzy logic, possibility theory and higher order generalizations) seem to fill the gap caused by the restricted frequency interpretation. Close examination shows that the fuzzy approaches have exactly the same representation as the corresponding probabilistc approach and include similar calculi. The probabilistic approach assumes less information when the calculi differ.
TL;DR: A fuzzy rule based expert production system that accepts as input a fuzzy vector all of whose components are fuzzy sets, and produces as output a fuzzy set of conclusions.
TL;DR: Several theorems which extend the possible effective application of Orlovsky's concept of decision-making on a finite set of alternatives with a fuzzy preference relation for optimization of many decision problems are formulated and proved.
TL;DR: It is shown that all fuzzy aggregation rules which have non-narrow domains and which satisfy the fuzzy counterparts of independence of irrelevant alternatives and Pareto criterion are characterized by a distribution of 'veto' power which would be generally considered undesirable.
TL;DR: Some aspects of manipulation of fuzzy data with the aid of fuzzy relational equations are considered and two stages of manipulation process are indicated: combining pieces of evidence and inferring their mutual correspondence.
01 Jan 1986
TL;DR: In this article, the authors described the application of the self-organizing fuzzy logic controller (SOC) to a complex multi-variable process and demonstrated that reasonable control can be obtained for this process.
Abstract: A study is described of the application of the self-organizing fuzzy logic controller (SOC), proposed by Procyk and Mamdani, to a complex multi-variable process. The control problem used in the investigation is the attitude control of a flexible satellite that has significant dynamic coupling of the axes. It is found that reasonable control can be obtained for this process, therefore demonstrating the potential of the SOC for complex processes that cannot be reliably modelled.
TL;DR: It is shown that if the notion of fuzzy sets is further fuzzified by making equality (as well as membership) fuzzy, the resultant categories are indeed toposes.
Abstract: The relation between the categories of Fuzzy Sets and that of Sheaves is explored and the precise connection between them is expli cated. In particular, it is shown that if the notion of fuzzy sets is further fuzzified by making equality (as well as membership) fuzzy, the resultant categories are indeed toposes.
TL;DR: A surprising result is that fuzzy functions take their values from the complemented elements of a Brouwerian lattice, suggesting that fuzzy set theory is more than just a generalization of ordinary set theory.
TL;DR: A new signal restoration method with considerable flexibility in incorporating a priori information is developed, which generated successful results in many restorations for which the conventional techniques have failed, and may be applied in image coding and tomography.
Abstract: A new signal restoration method with considerable flexibility in incorporating a priori information is developed. The method defines a fuzzy set for each piece of information to restrict the set of acceptable solutions. Using fuzzy sets makes it possible to model partially defined information as well as exact knowledge. The intersection of all the fuzzy sets is the feasibility set. The original signal is a member of this set with a high membership value, and any high membership valued element of this set is a nonrejectable solution. Such solutions can be computed by using optimization techniques. Ideally, the feasibility set contains only the original signal. The chance of recovering the original signal decreases as the feasibility set gets larger. Thus, the size of the feasibility set gives a quality measure for the solution. The method generated successful results in many restorations for which the conventional techniques have failed, and may be applied in image coding and tomography.
01 Mar 1986
TL;DR: An algorithm of ordering retrieved documents according to the values of the fuzzy thesaurus is proposed and it is shown that one can obtain documents of maximum relevance in a fixed time interval.
Abstract: A fuzzy bibliographic information retrieval based on a fuzzy thesaurus or on a fuzzy pseudothesaurus is described. A fuzzy thesaurus consists of two fuzzy relations defined on a set of keywords for the bibliography. The fuzzy relations are generated based on a fuzzy set model, which describes association of a keyword to its concepts. If the set of concepts in the fuzzy set model is replaced by the set of documents, the fuzzy relations are called a pseudothesaurus, which is automatically generated by using occurrence frequencies of the keywords in the set of documents. The fuzzy retrieval uses two fuzzy relations in addition, that is, a fuzzy indexing and a fuzzy inverted file: the latter is the inverse relation of the former. They are, however, related to different algorithms for indexing and retrieval, respectively. An algorithm of ordering retrieved documents according to the values of the fuzzy thesaurus is proposed. This method of the ordering is optimal in the sense that one can obtain documents of maximum relevance in a fixed time interval. An example of the fuzzy retrieval is shown on a prototype database. This method shows one of the simplest way to realize fuzzy retrieval in practical database systems.
TL;DR: In this article, the problem of the reconstruction of a fuzzy topological space or a fuzzy neighbourhood space from an a priori given family of level-topologies is discussed, and necessary and sufficient conditions for the existence of a solution are given.
TL;DR: An extension of the conventional dynamic programming model introduced by Bellman to the fuzzy case is considered and two fuzzy dynamic programming models are developed and converted into algo r performance of these algorithms is compared to two others based on heuristics.
TL;DR: An experimental construction of fuzzy set membership leads to a realization of stochastic fuzziness and a type II fuzzy set representation and axioms of measurement can be validated with a probabilistic interpretation.
Abstract: Empirical measurement of membership functions of fuzzy sets are considered with the fundamental axioms of measurement theory. An experimental construction of fuzzy set membership leads to a realization of stochastic fuzziness and a type II fuzzy set representation. Axioms of measurement can be validated with a probabilistic interpretation.